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Introduction To Options: Article 4 

Many times it is in an investor's best interest to lock in recent gains or to protect a portfolio of stocks from a decline beyond a certain price. One way to do this would be to purchase a put option contract on each of your various holdings (this would essentially allow you to "lock in" a particular sale price on each stock, so even if the market crashed, your overall portfolio wouldn't suffer much). However, if you hold a large, diversified portfolio of stocks, then it is probably not costeffective to insure each and every position in this manner.
As an alternative, you might want to consider using index options to hedge the risk in your portfolio. Index options are options not on an individual stock, but rather on an entire index. Many different indices have options available, including the Nasdaq 100, the Dow Jones Industrial Average and the S&P; 500. For the purposes of today's example we will use the S&P; 500 ticker symbol "SPX"as a proxy for the overall market's return. With some careful planning, you should be able to offset a sharp decline in your portfolio by hedging your overall position with index options. Though it is impossible to forecast exactly how your portfolio will perform during a steep market selloff, you can calculate this out fairly close to the actual result.
Before you can hedge your portfolio against a major market correction, however, you'll need to figure out two key items. First, you'll need to determine which particular index to use as a proxy for your portfolio. If you hold primarily hightech stocks (or if you just want to hedge against a downfall in your technology holdings), then you might want to consider trading options on the Nasdaq 100. Alternatively, if your portfolio consists mainly of bluechip companies, then you might want to use the Dow Jones Industrial Average. Again, since we're going to assume that your portfolio consists of a welldiversified mix of different stocks, for the purposes of today's example we'll use the S&P; 500 as our proxy.
Next, you'll need to find the correct number of options to use as a portfolio hedge. Along those lines, here are a few important items to consider:
 You first need to derive an estimate of beta ß. This may sound like an obscure technical term, but beta simply measures the amount of variance in a portfolio in relation to the market. If you were using the S&P; 500 as a proxy for the market, then ß would indicate how much your portfolio moves when the S&P; 500 changes by 1%. For example, if you notice that, in general, your portfolio changes by 2% whenever the S&P; moves up 1%, then your portfolio has a ß of 2.0. If the portfolio changes by 0.5%, then ß = 0.5. If the portfolio changes by 1%, then ß = 1.0. (Beta is an important component of all options, so it would be a useful exercise to try this with your portfolio or individual stocks to become more comfortable with this term.)
 The next step is finding the riskfree rate. As the name implies, this is the rate of interest that can be obtained without incurring any risk. For the shortterm, we usually use the appropriate threemonth Tbill rate.
If your portfolio pays any dividends, then you need to formulate the portfolio's dividend yield. This can be found by adding the amount of dividends paid during the year and dividing that figure by the value of your portfolio. For example, if you receive roughly $40,000 in dividends per year on a $1 million portfolio, then your portfolio's dividend yield is 4% (40,000 ÷ 1,000,000).
Now consider the following example:
Suppose you own a $1 million portfolio of stocks and you wish to insure this portfolio against a decline of greater than 6% during the next three months. In other words, you want to put a hedge in place to make certain that your portfolio does not fall below $940,000. To make the calculations fairly simple, let's assume that the S&P; 500 index is currently trading at 1000. Let's also assume that your portfolio is volatile and generally doubles the S&P; 500's gains or losses. Therefore, the ? is 2.0. Finally, let's assume that the riskfree rate is 4% and the dividend yields on both your portfolio and the S&P; 500 are also 4%. The assumed return of the SPX is 12% per year. (We should note that it is not necessary to have an accurate forecast of the market's return for this hedge to work correctly.) In this example, if you want to employ SPX put options as a hedging tool, then here's how to calculate how many contracts you need to purchase:
Total Return of SPX in Three Months:
In three month's time you expect a 3% return (assuming a 12% annual rate) and a 1% dividend (assuming a 4% annual yield) for a total return of 4%.
Excess Return of SPX:
The excess return of any asset is the amount it returns over the riskfree rate. In this case, the riskfree rate would be 1% (assuming a 4% annual rate) in three months. The excess return is therefore 3% (4% total return  1% riskfree).
Total Return of Portfolio in Three months:
For this example we stated that the ? of the portfolio is 2, which implies that if the market returns 3%, then your portfolio will double that amount by returning 6%. The expected dividend is still 1% during the next three months, so the total expected return will be 7%.
Excess Return of Portfolio:
The expected excess return is 7% and the riskfree rate is 1%, so the excess return here will be 6%. This is the return you expect in three month's time.
The table below illustrates how the portfolio is expected to behave in relation to the market:
Value of S&P; 500 Index in Three Months 
Value of Portfolio in Three Months 
1060 
$1,120,000 
1030 
$1,060,000 
1000 
$1,000,000 
970 
$940,000 
940 
$880,000 
From this chart we can see that the portfolio will perform twice as well, or twice as poorly, as the market. In this example, you do not want to let your portfolio fall below $940,000 in the next three months. Using the table, you can see that buying SPX puts with a strike price of 970 will accomplish this. To find the optimal number of put option contracts to purchase, use this formula:
Portfolio Value ÷ [(100 x Current Strike Price) ÷ ß] = number of put contracts
In this example, where the portfolio value equals $1,000,000 and the current strike price is 1000, the calculation would be as follows:
$1,000,000 ÷ [(100,000) ÷ 2] = $1,000,000 ÷ 50,000 = 20 put contracts
This means you should buy 20 SPX 970 Put contracts that expire in three months to insure your portfolio against a decline below $940,000.
To see that this is correct, suppose the SPX finishes at 940 when the options expire in three months. This implies from the chart that your portfolio would be worth just $880,000. However, the SPX 970 Put contract will expire "in the money" and will be worth $30 (970  940) at expiration. In this scenario, your 20 options contracts (each contract is for 100 options) will now carry a value of $60,000. When you add that figure to the $880,000 that your portfolio is now worth, this equals exactly $940,000. You can perform this procedure for any decline in the SPX, and in every case you will find that your overall portfolio will still be worth a minimum of $940,000 at expiration.
Conclusion
Portfolio hedging is an important technique to learn. Although the calculation of ? must be correct to ensure an exact result, most investors find that even a reasonable approximation will deliver a satisfactory hedge. This technique is especially helpful when an investor has experienced an extended period of gains and feels this increase might not be sustainable in the future. Like all option strategies, portfolio hedging requires a little planning before executing a trade. However, the security that this strategy provides could make it well worth the time and effort in a period of declining stock prices.




